FLOW PAST A VERTICAL POROUS PLATE
M. GURIA AND R. N. JANA Received 11 April 2005
The study of unsteady hydrodynamic free convective flow of a viscous incompressible fluid past a vertical porous plate in the presence of a variable suction has been made.
Approximate solutions have been derived for the velocity and temperature fields, shear stress, and rate of heat transfer using perturbation technique. It is observed that main fluid velocity decreases with increase in Prandtl number, while it increases with increase in suction parameter. The cross-velocity decreases near the plate and increases away from the plate with increase in suction parameter. On the other hand, it increases near the plate and decreases away from the plate with increase in frequency parameter. The amplitude and the tangent of phase shift of the shear stress due to main flow decrease with increase in either Prandtl number, Grashof number, or frequency parameter. It is seen that the temperature decreases with increase in either suction parameter, Prandtl number, or fre- quency parameter. It is also seen that the amplitude of the rate of heat transfer increases and the tangent of phase shift of rate of heat transfer decreases with increase in Prandtl number.
1. Introduction
The research area of laminar flow is continuously growing, and it is the subject of in- tensive studies in recent years because of its application in engineering, particularly in aeronautical engineering. One of the most important application of laminar flow is the calculation of friction drag of bodies in a flow, for example, the drag of a plate at zero in- cidence, the friction drag of ship, an airfoil. It is also important for heat transfer between a body and the fluid around it. The effect of different arrangements and configurations of the suction holes and slits on the drag has been studied by various scholars. Misra et al. [4] have studied the effect of buoyancy forces on the three-dimensional flow and heat transfer along a porous vertical plate. Misra et al. [3] also studied the flow of viscous incompressible fluid along an infinite porous plate by applying the transverse sinusoidal suction velocity distribution fluctuating with time. Later, Singh [2] extended this idea by applying transverse sinusoidal suction velocity in the presence of viscous dissipative heat.
Copyright©2005 Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences 2005:20 (2005) 3359–3372 DOI:10.1155/IJMMS.2005.3359
Singh [1] also discussed the effect of magnetic field on the three-dimensional flow past a porous plate.
The aim of this note is to study the effect of buoyancy forces and time-dependent pe- riodic suction on three-dimensional flow past a vertical porous plate. The velocity field, shear stress, temperature distribution, and the rate of heat transfer have been derived. It is observed that main fluid velocity decreases with increase in Prandtl number while it increases with increase in suction parameter. The cross-velocity decreases near the plate and increases away from the plate with increase in suction parameter. On the other hand, it increases near the plate and decreases away from the plate with increase in frequency parameter. The amplitude and the tangent of phase shift of the shear stress due to main flow decrease with increase in either Prandtl number, Grashof number, or frequency pa- rameter. It is seen that the temperature decreases with increase in either Prandtl number, suction parameter, or frequency parameter. It is also seen that the amplitude of the rate of heat transfer increases and the tangent of phase shift of rate of heat transfer decreases with increase in Prandtl number.
2. Formulation of the problem
Consider the unsteady flow of viscous, incompressible fluid past along a semi-infinite vertical porous plate. Here, thex-axis is chosen along the vertical plate, that is, the di- rection of the flow, y-axis is perpendicular to the plate, andz-axis is normal to the xy-plane.
The plate is subjected to periodic suction velocity distribution of the form
v= −V0
1 +cos
πu∞z
ν −ct, (2.1)
where(1) is the amplitude of the suction velocity. Denoting velocity components u,v,win the directionsx-, y-,z-axes, respectively, the flow is governed by the following equations:
∂v
∂y+∂w
∂z =0, (2.2)
∂u
∂t +v∂u
∂y+w∂u
∂z =ν∂2u
∂y2+∂2u
∂z2
+gβT−T∞
, (2.3)
∂v
∂t+v∂v
∂y+w∂v
∂z = − 1 ρ
∂p
∂y+ν∂2v
∂y2+∂2v
∂z2
, (2.4)
∂w
∂t +v∂w
∂y+w∂w
∂z = − 1 ρ
∂p
∂z+ν∂2w
∂y2 +∂2w
∂z2
, (2.5)
∂T
∂t+v∂T
∂y+w∂T
∂z= K ρCp
∂2T
∂y2+ ∂2T
∂z2
, (2.6)
whereρis the density, pis the fluid pressure,g is the acceleration due to gravity, the coefficient of thermal expansion isβ, the coefficient of heat conduction isK, the specific heat at constant pressure isCp.
The boundary conditions of the problem are u=0, v= −V0
1 +cos
πu∞z
ν −ct, w=0, T=Tw aty=0, u=u∞, v= −V0, w=0, p=p∞, T=T∞ aty= ∞.
(2.7) Introduce the nondimensional variables
y=u∞y
ν , z=u∞z
∞ , t=ct, p= p ρu2∞, u=u
u∞, v=v
u∞, w=w u∞, θ=
T−T∞ Tw−T∞.
(2.8)
Using (2.8), (2.2)–(2.6) become
∂v
∂y+∂w
∂z =0, (2.9)
ω∂u
∂t +v∂u
∂y+w∂u
∂z=∂2u
∂y2+∂2u
∂z2 +Grθ, (2.10)
ω∂v
∂t +v∂v
∂y+w∂v
∂z= −
∂p
∂y + ∂2v
∂y2+∂2v
∂z2
, (2.11)
ω∂w
∂t +v∂w
∂y +w∂w
∂z = −
∂p
∂z + ∂2w
∂y2 +∂2w
∂z2
, (2.12)
ω∂θ
∂t +v∂θ
∂y+w∂θ
∂z= 1 Pr
∂2θ
∂y2+∂2θ
∂z2
, (2.13)
where Gr=gβ(Tw−T∞)ν/u3∞is the Grashof number,ω=cν/u2∞is the frequency param- eter, and Pr=ρνCp/Kis the Prandtl number.T∞is the temperature outside the boundary layer,p∞is pressure outside the boundary layer.
The boundary conditions (2.7) become
u=0, w=0, v= −S1 +cos(πz−t), θ=1 aty=0,
u=1, w=0, v= −S, θ=0 aty−→ ∞, (2.14) whereS=V0/u∞is the suction parameter.
3. Solution of the problem
To solve (2.9)–(2.13), we assume the solution of the following form:
u=u0+u1+2u2+···, v=v0+v1+2v2+···, w=w0+w1+2w2+···,
p=p0+p1+2p2+···, θ=θ0+θ1+2θ2+···.
(3.1)
Substituting (3.1) in (2.9)–(2.13), comparing the term free fromand the coefficient of from both sides, and neglecting those of2, the term free fromis
v0=0, (3.2)
u0 −v0u0+ Grθ0=0, (3.3)
θ0 −v0Prθ0=0, (3.4)
where the primes denote differentiation with respect toy.
The boundary conditions are
u0=0, v0= −S, θ0=1 aty=0,
u0=1, v0= −S, θ0=0 aty−→ ∞. (3.5) The solutions of (3.2)–(3.4) under the boundary conditions (3.5) are
v0(y)= −S, θ0(y)=e−SPry, u0(y)=
1−e−Sy− −Gr S2Pr(Pr−1.0)
e−SPry−e−Sy for Pr=1,
=1−e−Sy+Gr
S ye−Sy for Pr=1.0.
(3.6)
Equating the coefficient offrom both sides, we get
∂v1
∂y +∂w1
∂z =0, (3.7)
ω∂u1
∂t +v1∂u0
∂y −S∂u1
∂y =
∂2u1
∂y2 +∂2u1
∂z2 + Grθ1, (3.8)
ω∂v1
∂t −S∂v1
∂y = −
∂p1
∂y + ∂2v1
∂y2 +∂2v1
∂z2
, (3.9)
ω∂w1
∂t −S∂w1
∂y = −
∂p1
∂z + ∂2w1
∂y2 +∂2w1
∂z2
, (3.10)
ω∂θ1
∂t +v1∂θ0
∂y −S∂θ1
∂y = 1 Pr
∂2θ1
∂y2 +∂2θ1
∂z2
. (3.11)
The boundary conditions become
u1=0, v1= −Scos(πz−t), w1=0, θ1=0 aty=0,
u1=0, v1=0, w1=0, θ1=0, p1=0 aty−→ ∞. (3.12) These are the linear partial differential equations describing the three-dimensional flow.
We assume the velocity components, pressure, and temperature in the following form:
u1(y,z,t)=u11(y)ei(πz−t), v1(y,z,t)=v11(y)ei(πz−t), w1(y,z,t)= i
πv11(y)ei(πz−t), p1(y,z,t)=p11(y)ei(πz−t), θ1(y,z,t)=θ11(y)ei(πz−t).
(3.13)
Substituting (3.13) in (3.7)–(3.11), we get the following set of differential equations:
v11+Sv11 −
π2−iωv11=p11 , (3.14)
v11+Sv11−
π2−iωv11=π2p11, (3.15) θ11+SPrθ11−
π2−iPrωθ11=Prv11θ0, (3.16) u11+Su11−
π2−iωu11=v11u0−Grθ11. (3.17) The boundary conditions become
u11=0, v11= −S, w11=0, θ11=0 aty=0,
u11=0, v11=0, w11=0, θ11=0 aty−→ ∞. (3.18) Solving (3.14)–(3.17), under the boundary conditions (3.18), we get
v1(y,z)= S π−r1
r1e−π y−πe−r1yei(πz−t), w1(y,z)= iSr1
π−r1
e−r1y−e−π yei(πz−t), p1(y,z)= Sr1
ππ−r1
(Sπ−iω)e−π yei(πz−t), θ1(y,z)=S2Pr2
π−r1
Ce−r2y− r1
Pr(Sπ+iω)e−(π+SPr)y
+ π
Sr1(1 + Pr) +iω(Pr−1.0)e−(r1+SPr)yei(πz−t),
u1(y,z)= S2 π−r1
De−r1y+K1e−(π+S)y+K2e−r2y+K3e−(r1+S)y
+K4e−(π+SPr)y+K5e−(r1+SPr)yei(πz−t) for Pr=1.0,
= S π−r1
−
C2+C3+C4+C5+GrSBy S−2r1
e−r1y +
C2+C4− r1Gry Sπ+iω
e−(π+S)y+
C3+C5+πGry 2Sr1
e−(r1+S)y
×ei(πz−t) for Pr=1.0,
(3.19) where
r1=S+S2+ 4π2−iω
2 , r2=SPr +S2Pr2+4π2−iωPr
2 ,
C= r1
Pr(Sπ+iω)− π
Sr1(Pr +1) +iω(Pr−1), K1=r1
1 +C1
(Sπ+iω), K2= −Gr Pr2C
r22−Sr2−π2+iω, K3=−π1 +C1
2Sr1 , K4= Gr Pr2r1
Pr(Sπ+iω)−r1C1PrS2Pr(Pr−1) +Sπ(2 Pr−1.0) +iω, K5=
πC1Pr− πGr Pr2 Sr1(Pr +1) +iω(Pr−1)
SPr2r1+SPr−S,
C1= −Gr
S2Pr(Pr−1.0), C2= r1
S+Gr S
+ GrSr1
Sπ+iω 1
Sπ+iω, C3= −
πS+Gr S
+Grπ
2r1
1 2Sr1
, C4=−Grr1(2π+S)
(Sπ+iω)2 , C5=πGrS+ 2r1
4S2r12
. (3.20) 4. Result and discussion
We have presented the nondimensional main flow velocityuand cross-velocitywagainst yfor different values of Pr,S, andωand for Gr=5.0,z=0.2,t=0.2. The main flow velocity profiles are shown graphically in Figures4.1and4.2. We observed that the main flow decreases with increase in Prandtl number while it increases with increase in suction parameter. Also we obtainedufor different values ofωwhich is given inTable 4.1. From the table, it is seen thatuincreases with increase inω, but the effect is negligible. From Figure 4.3, it is seen that the cross-flow decreases near the plate and increases away from the plate with increase in suction parameter.Figure 4.4shows that the magnitude of the cross-flow increases near the plate and decreases away from the plate with increase in frequency parameterω.
The main velocityufor different values ofωis shown in the table.
1 0.8
0.6
0.4
0.2
0 u
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
y Pr=1
Pr=3 Pr=4
Pr=5 Pr=6
Figure 4.1. Main flow velocityufor Gr=5.0,S=1.0,ω=10,z=0.2,t=0.2,=0.2.
1
0.8
0.6
0.4
0.2
0 u
0 1 2 3 4 5 6 7 8 9 10
y S=0.5
S=0.7
S=0.9 S=1
Figure 4.2. Main flow velocityufor Gr=5.0, Pr=7.0,ω=10,z=0.2,t=0.2,=0.2.
The important characteristic of the problem is shear stress. The shear stress due to main flow direction at the platey=0 is
τx= ∂u
∂y
y=0=u0(0) +u1(0)=u0(0) +u11(0)ei(πz−t). (4.1)
Table 4.1. Main velocityufor Gr=5.0, Pr=7.0,S=1.0, andz=0.2.
y u
ω=5 ω=8 ω=10 ω=15 ω=20
0.00 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 1.00 0.67414890 0.67491380 0.67526410 0.67564240 0.67575110 2.00 0.88072730 0.88076850 0.88077700 0.88077790 0.880777640 3.00 0.95613920 0.95614020 0.95614010 0.95614000 0.95614000 4.00 0.98386490 0.98386490 0.98386490 0.98386490 0.98386490 5.00 0.99406430 0.99406430 0.99406430 0.99406430 0.99405430 6.00 0.99781640 0.99781640 0.99781640 0.99781640 0.99781640 7.00 0.99919670 0.99919670 0.99919670 0.99919670 0.99919670 8.00 0.99970450 0.99970450 0.99970450 0.99970450 0.99970450 9.00 0.99989130 0.99989130 0.99989130 0.99989130 0.99989130 10.00 0.99996010 0.99996010 0.99996010 0.99996010 0.99996010
0.2 0
−0.2
−0.4
−0.6
−0.8
−1
−1.2
−1.4
−1.6
−1.8
10w
0 0.5 1 1.5 2 2.5 3
y S=0.1
S=0.3 S=0.5
S=0.7 S=0.9 S=1
Figure 4.3. Cross-velocity 10×wfor Gr=5.0,ω=10,z=0.2,t=0.2,=0.2.
We express the shear stress component in terms of magnitude and tangent of phase shift:
τx=u0(0) +R1cosπz−t+φ1
for Pr=1.0,
τx=u0(0) +R1cosπz−t+φ1 for Pr=1.0, (4.2)
2
1.5
1
0.5
0
−0.5
−10w
0 0.5 1 1.5 2 2.5 3
y ω=2
ω=5 ω=8
ω=10 ω=15
Figure 4.4. Cross-velocity−10×wfor Gr=5.0,S=1.0,z=0.2,t=0.2,=0.2.
where R1=
u21r+u21i, tanφ1= u1i
u1r, R2=
u211r+u211i, tanφ2=u11i
u11r. (4.3) The magnitude and the tangent of phase shift of the shear stress due to main flow are shown graphically in Figures4.5,4.6,4.7, and4.8againstωfor different values of Prandtl number and Grashoffnumber. It is seen that the magnitude and the tangent of phase shift of the shear stress decrease with increase in either Prandtl number Pr, Grashof number Gr, or frequency parameterω.
The temperatureθfor different values ofωis shown in the table.
The temperature distribution has been obtained and plotted for different values of Pr andSin Figures4.9and4.10forω=10.0, Gr=5.0,t=0.2. It is found that the temper- atureθdecreases with increase in either Prandtl number or suction parameter. Also we have obtained the temperature distribution for different values of frequency parameter which is given inTable 4.2. From the table, it is found that the temperature decreases with increase in frequency parameter.
Also, we calculate the rate of heat transfer. The rate of heat transfer at the platey=0 is given by
∂θ11
∂y
y=0=θ0(0) +θ1(0)
=θ0(0) +θ11 (0)ei(πz−t)
=θ0(0) +R3cosπz−t+φ3
.
(4.4)
1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2
|R1|
0 6 7 8 9 10 11 12 13 14 15
ω Pr=2
Pr=2.5 Pr=3
Pr=5 Pr=15
Figure 4.5. Amplitude of the shear stress due to main flow for Gr=5.0,S=1.0,z=0.2.
2.5 2 1.5 1 0.5 0
−0.5
−1 tanφ1
5 6 7 8 9 10 11 12 13 14 15
ω Pr=2
Pr=3 Pr=4
Pr=5 Pr=6
Figure 4.6. Tangent of phase shift of shear stress due to main flow for Gr=5.0,S=1.0,z=0.0.
We draw the graph of amplitude and tangent of phase shift of the rate of heat transfer against frequency parameterωfor different values of Prandtl number Pr in Figures4.11
0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3
|R2|
5 6 7 8 9 10 11 12 13 14 15
ω Gr=1
Gr=2 Gr=3
Gr=4 Gr=5
Figure 4.7. Amplitude of the shear stress due to main flow forS=1.0.
1 0.8 0.6
0.4 0.2
0
−tanφ2
5 6 7 8 9 10 11 12 13 14 15
ω Gr=1
Gr=2 Gr=3
Gr=4 Gr=5 Gr=6
Figure 4.8. Tangent of phase shift of the shear stress due to main flow forS=1.0.
and4.12and for Gr=5.0,S=1.0. FromFigure 4.11, we see that the amplitude increases with increase in Prandtl number but decreases with increase in frequency parameterω.
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 θ
0 0.5 1 1.5 2 2.5 3
y Pr=2
Pr=2.5 Pr=3
Pr=4 Pr=5
Figure 4.9. Temperature profile for Gr=5.0,S=1.0,ω=10.0,z=0.2,t=0.2,=0.2.
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 θ
0 1 2 3 4 5 6 7 8 9 10
y S=0.3
S=0.5 S=0.7
S=0.9 S=1
Figure 4.10. Temperature profile for Gr=5.0, Pr=2.0,ω=10.0,z=0.2,t=0.2,=0.2.
Figure 4.12shows that the magnitude of tangent of phase shift decreases with increase in Prandtl number Pr but increases with increase inω. It is seen that there is always a phase lag.
Table 4.2. Temperature distribution for Gr=5.0,S=1.0, Pr=2.0.
y θ
ω=5 ω=8 ω=10 ω=15 ω=20
0.00 1.00000000 1.00000000 1.00000000 1.00000000 1.00000000 0.40 0.46545780 0.45846200 0.45615440 0.45372500 0.45261860 0.80 0.20575920 0.20351230 0.20298460 0.20245720 0.20221060 1.20 0.09137923 0.09091454 0.09083322 0.09075849 0.09073284 1.60 0.04085174 0.04077750 0.04077017 0.04076410 0.04076333 2.00 0.01832459 0.01831620 0.01831609 0.01831586 0.01831588 2.40 0.00823015 0.00822975 0.00822981 0.00822979 0.00822978 2.80 0.00369778 0.00369788 0.003697888 0.00369787 0.00369787 3.20 0.00166153 0.00166156 0.00166156 0.00166156 0.00166156 3.60 0.00074658 0.00074659 0.00074659 0.0074659 0.00074659 4.00 0.00033546 0.00033546 0.00033546 0.00033546 0.00033546
6 5
4 3
2 1 0
|R3|
5 6 7 8 9 10 11 12 13 14 15
ω Pr=2
Pr=3 Pr=4
Pr=5 Pr=6
Figure 4.11. Amplitude of the rate of heat transfer for Gr=5.0,S=1.0.
Acknowledgment
M. Guria is highly thankful to the University Grants Commission (UGC) for granting the fellowship.
1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2
−tanφ3
0 6 7 8 9 10 11 12 13 14 15
ω Pr=2
Pr=3 Pr=4
Pr=5 Pr=6
Figure 4.12. Tangent of phase shift of the rate of heat transfer for Gr=5.0,S=1.0.
References
[1] K. D. Singh,Hydromagnetic effects on the three-dimensional flow past a porous plate, Z. Angew.
Math. Phys.41(1990), no. 3, 441–446.
[2] ,Three-dimensional viscous flow and heat transfer along a porous plate, Z. Angew. Math.
Mech.73(1993), no. 1, 58–61.
[3] P. Singh, V. P. Sharma, and U. N. Misra,Three dimensional fluctuating flow and heat transfer along a plate with suction, Int. J. Heat Mass Transfer21(1978), 1117–1123.
[4] ,Three dimensional free convection flow and heat transfer along a porous vertical plate, Appl. Sci. Res.34(1978), no. 1, 105–115.
M. Guria: Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721 102, West Bengal, India
E-mail address:[email protected]
R. N. Jana: Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721 102, West Bengal, India
E-mail address:[email protected]
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